In some state, there are 101 cities.
a) Each city is connected to each of the other cities by one-way roads, and 50 roads lead into each city and 50 roads lead out of each city. Prove that you can get from each city to any other, having travelled on no more than on two roads.
b) Some cities are connected by one-way roads, and 40 roads lead into each city and 40 roads lead out of each. Prove that you can get form each city to any other, having travelled on no more than on three roads.
In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.
In what number system is the equality \(3 \times 4 = 10\) correct?
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(a, b, c > 0\), the following inequality is valid: \(\left(\frac{a+b+c}{3}\right)^2 \ge \frac{ab+bc+ca}{3}\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).
Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
Prove that any axis of symmetry of a 45-gon passes through its vertex.
Is the number \(1 + 2 + 3 + \dots + 1990\) odd or even?
Every Martian has three hands. Can seven Martians join hands?